U.S. patent number 5,243,672 [Application Number 07/924,772] was granted by the patent office on 1993-09-07 for planar waveguide having optimized bend.
This patent grant is currently assigned to AT&T Bell Laboratories. Invention is credited to Corrado Dragone.
United States Patent |
5,243,672 |
Dragone |
September 7, 1993 |
Planar waveguide having optimized bend
Abstract
In an embodiment, an N.times.N integrated optical
interconnection apparatus capable of switching, multiplexing or
demultiplexing a large number of input and output wavelength
channels achieves low levels of crosstalk and insertion loss. Two
substantially identical N.times.M star couplers are connected by an
optical diffraction grating comprising M unequal length waveguides
spaced from one another by predetermined amounts. The waveguides of
the grating consists of an array of curved waveguides of different
lengths. The waveguides are closely spaced at their ends and widely
spaced and strongly curved in the central region. The curves or
bends of the planar waveguides here disclosed have sharper bends,
lower losses and increased tolerance to fabrication defects. The
improved bends are realized by selecting a width and radius of
curvature for the waveguide which is large enough to force the
fundamental mode of an optical signal to propagate away from the
inner edge of the bend, thus causing negligible illumination of
this edge. Moreover, the curvature of the bend is large enough to
effectively cut off modes above the fundamental mode. In the new
waveguide here disclosed, the mode propagation constant is
effectively independent of the waveguide width thus avoiding prior
art phase error problems caused by variations in the width of the
waveguide.
Inventors: |
Dragone; Corrado (Little
Silver, NJ) |
Assignee: |
AT&T Bell Laboratories
(Murray Hill, NJ)
|
Family
ID: |
25450699 |
Appl.
No.: |
07/924,772 |
Filed: |
August 4, 1992 |
Current U.S.
Class: |
385/46; 385/17;
385/37; 385/39 |
Current CPC
Class: |
G02B
6/12011 (20130101); G02B 6/2804 (20130101); G02B
6/262 (20130101); G02B 6/125 (20130101) |
Current International
Class: |
G02B
6/26 (20060101); G02B 6/125 (20060101); G02B
6/28 (20060101); G02B 6/34 (20060101); G02B
006/26 () |
Field of
Search: |
;385/13,27,28,32,46,17,16,37,39,42,43,44,45 |
References Cited
[Referenced By]
U.S. Patent Documents
Other References
"Broadband Silica-Based Optical Waveguide Coupler with Asymmetric
Structure" A. Takagi, et al. Electronic Letters, Jan. 18, 1990, No.
2, vol. 26, pp. 132-133..
|
Primary Examiner: Gonzalez; Frank
Attorney, Agent or Firm: Weiss; Eli
Claims
I claim:
1. An optical interconnection apparatus comprising:
a first plurality of input waveguides radially directed from a
plurality of input ports toward a first focal point;
a first star coupler having an input connected to the plurality of
input waveguides;
a first plurality of output waveguides radially directed to a
second focal point and connected to an output of the first star
coupler;
an optical grating comprising a plurality of unequal length
waveguides having inputs connected to the first plurality of output
waveguides;
a second plurality of input waveguides radially directed to a third
focal point and connected to outputs of the optical grating;
said waveguides of said optical grating have a width w which is not
less than ##EQU7## where: .lambda. is the wavelength of the
fundamental mode of the signal in the waveguide
n is the effective refractive index of the core of the waveguide;
and
.DELTA.n.sub.e is the effective refractive index difference between
the core and the cladding; and a bend, the radius of which is sized
to effectively cut off modes above the fundamental mode of an
optical signal therein;
a second star coupler having an input connected to an output of the
second plurality of input waveguides; and
a second plurality of output waveguides radially directed from a
plurality of output ports toward a fourth focal point and connected
to an output of the second star coupler;
the first and second focal points being located predetermined
distances from the first star coupler and the third and fourth
focal points being located predetermined distances from the second
star coupler.
2. The optical interconnection apparatus of claim 1 wherein
said bend has a radius of curvature R which is not greater than
##EQU8## where: .lambda. is the wavelength of the fundamental mode
of the signal in the waveguide
n is the effective refractive index of the core of the waveguide;
and
.DELTA.n.sub.e is the effective refractive index difference between
the core and the cladding.
3. The optical interconnection apparatus of claim 2 wherein
said unequal length waveguides of said grating are planar
waveguides.
4. The optical interconnection apparatus of claim 2 wherein
the bend in the waveguide has a radius which causes a loss in the
bend due to tunneling to exceed 0.01 dB/radian.
5. The optical interconnection apparatus of claim 2 wherein
the bend in the waveguide is butt coupled to a straight waveguide,
the axis of the straight waveguide being aligned with the bend
mode.
6. The optical interconnection apparatus of claim 5 wherein the
axis of the straight waveguide is displaced from the outside edge
of the waveguide with the bend by the distance .delta..sub.g where
##EQU9## where R is the radius of the bend,
n is the refractive index of the core
.DELTA.n.sub.e is the refractive difference between the core and
the cladding;
.lambda. is waveguide wavelength; and
k=2.pi./.lambda.=6.283/.lambda..
7. The optical interconnection apparatus of claim 3 wherein
said waveguides of the grating are planar waveguides.
8. A planar waveguide for conducting an optical signal along a
curved path, the width of the curve path being large enough to
cause the fundamental mode of the optical signal to be displaced
from the inner edge of the curved path, and the curvature of the
curved path being sufficiently large to effectively cut off modes
above the fundamental mode of the optical signal.
9. The planar waveguide of claim 8 wherein the planar waveguide has
a width which is equal to or greater than ##EQU10## where .lambda.
is a waveguide wavelength,
n is the refractive index of the core, and
.DELTA.n.sub.e is the effective refractive index between the core
and the cladding.
10. The planar waveguide of claim 9 wherein the curved path has a
radius which is equal to or smaller than ##EQU11## where .lambda.
is the wavelength of the waveguide,
n is the refractive index of the core; and
.DELTA.n.sub.e is the effective refractive index between the core
and the cladding.
11. A method of coupling a straight waveguide to a curved planar
waveguide comprising the steps of
determining the distance .delta..sub.g of the bend mode of the
curved planar waveguide from the outside curved edge of the planar
waveguide where ##EQU12## where .lambda. is the wavelength of the
waveguide,
n is the refractive index of the core,
.DELTA.n.sub.e is the effective refractive index between the core
and the cladding; and
k equals 2(.pi./.lambda.,)
and
aligning the axis of the straight waveguide with the bend mode of
the curved planar waveguide.
Description
FIELD OF THE INVENTION
This invention relates generally to planar waveguides used to carry
optical signals between optical devices and/or between optical
devices and other waveguides. More particularly, this invention
relates to an improved bend in a planar waveguide which has a
smaller radius, lower losses and increased tolerance to fabrication
defects.
BACKGROUND OF THE INVENTION
Optical switching, multiplexing, and demultiplexing have been
accomplished in the past by using an interconnection apparatus
having a plurality of closely spaced input waveguides communicating
with the input of a star coupler. The output of the star coupler
communicates with a second star coupler via an optical grating
consisting of an array of optical waveguides. Each of the
waveguides differs in length with respect to its nearest neighbor
by a predetermined fixed amount. The outputs of the second star
coupler form the outputs of the switching, multiplexing, and
demultiplexing apparatus. See, for example, my U.S. Pat. No.
5,002,350 issued Mar. 26, 1991.
In operation, when each of a plurality of separate and distinct
wavelengths are launched into a separate and distinct input port of
the apparatus, they will all combine and appear on a predetermined
one of the output ports. In this manner, the apparatus performs a
multiplexing function. The same apparatus may also perform a
demultiplexing function. In this situation, a plurality of input
wavelengths is directed to a predetermined one of the input ports
of the apparatus. Each of the input wavelengths is separated from
the others and directed to a predetermined one of the output ports
of the apparatus. An appropriate selection of input wavelength also
permits switching between any selected input port to any selected
output port.
The grating located between the two star couplers essentially
consists of an array of curved waveguides of different lengths. The
waveguides are closely spaced at their ends, whereas they are
widely spaced and strongly curved in the central region. The order
of the grating is determined by the difference in length between
adjacent waveguides. For many applications, the order of the
grating must be large, normally greater than 50. As a result, the
grating then becomes large and is difficult to make with
satisfactory accuracy, particularly when very low levels of
cross-talk are desired. Actually, defects of fabrication will, in
general, cause waveguide width variations which will affect the
propagation constant in each arm of the grating, thus causing phase
errors that will substantially increase cross-talk in a
multiplexer. In addition, to keep bend losses to a minimum, the
radius of the waveguide bend is limited.
It is, therefore, an object of this invention to reduce phase
errors in grating waveguides which result from fabrication
variations. It is also an object of this invention to reduce the
bend loss to allow the bend radius of the waveguide bend to be
further decreased without further increasing bend losses.
SUMMARY OF THE INVENTION
This object is achieved by making the curvature of a planar
waveguide to be as close to the critical value needed to insure
that essentially only the fundamental mode propagates with the
largest loss that can be tolerated. Also, the width of the planar
waveguide is made sufficiently large to cause the fundamental mode
of the optical signal to be displaced away from the inner edge of
the curve and, therefore, to be concentrated along the outside edge
of the curve. Thus, the fundamental mode effectively propagates in
the vicinity of the outer edge of the bend and its propagation
constant becomes effectively independent of the width of the
waveguide. As a result, width variations of the planar waveguide
caused by fabrication errors do not contribute to loss and, as the
field intensity is small at the inner edge of the bend, negligible
loss is caused by scattering from this edge.
BRIEF DESCRIPTION OF THE DRAWING
In the drawing:
FIG. 1 illustrates an example of an integrated optical switching,
multiplexing, and demultiplexing apparatus in which the invention
can be used;
FIG. 2 depicts one-half of one of the waveguides in the optical
grating shown in FIG. 1;
FIG. 3 illustrates in greater detail one-half of the optical
grating of FIG. 1; and
FIG. 4 illustrates the coupling relationship of the straight
waveguide with the bend of a planar waveguide in accordance with
the principles of the invention.
DETAILED DESCRIPTION
FIG. 1 shows an example of an optical interconnection apparatus
which can be used as an optical switch, multiplexer, or
demultiplexer in accordance with this invention. It preferably
comprises two substantially identical and symmetrically disposed
star couplers 10 and 12 connected by waveguides forming a
substantially symmetrical optical diffraction grating 14.
In FIG. 1, an array 16 of N input waveguides are radially directed
from N input ports toward a focal point F2. Each of the input
waveguides has a predetermined width W and is angularly displaced
from its adjacent waveguides by an angle .alpha..
The star coupler 10 comprises a dielectric slab 18 which forms a
free space region having two curved, preferably circular,
boundaries 18a and 18b. The input waveguides in the array 16 are
connected to the free space region 18 in a substantially uniform
fashion along boundary 18a. As indicated in FIG. 1, each of the
waveguides is separated from its neighbor by a distance t along the
boundary 18a.
An array 20 of M output waveguides is radially directed toward a
focal point F1. Each of the waveguides in the array 20 has a width
W' and is separated from adjacent waveguides in the array 20 by an
angular spacing .alpha.'. The output waveguides in the array 20 are
connected to the free space region 18 in a substantially uniform
fashion along boundary 18b. Each of the output waveguides is
separated from its neighbors at the boundary 18b by a distance t',
as shown in FIG. 1.
The M waveguides of the grating 14 are a symmetric arrangement of
waveguides each having lengths l.sub.s, where s is referenced to
the central waveguide in the grating. Each half of the grating 14
comprises preferably three sections, respectively composed of
radial, circular, and equispaced parallel waveguides. The total
length of the s.sup.th waveguide is
where h.sub.o is a constant and R.sub.s is the s-th radius of
curvature.
Each of the output waveguides in the array 20 is connected to the
input of a waveguide in the grating 14. The length of each
waveguide in the grating differs from the lengths of all the other
waveguides in the grating so that, in effect, predetermined and
different phase shifts are applied to optical signals propagating
into the waveguides of the grating from the star coupler 10 because
of the different path lengths over which the signals in the grating
must travel to reach the output of the grating. The outputs of the
waveguides in the grating 14 thus have different phases, which are
functions of the lengths of the waveguides.
The outputs of the waveguides in the grating 14 are connected to
another array 22 of M input waveguides which are radially directed
toward a focal point F4. The array 22 connects the output of the
grating 14 to the input of the second star coupler 12. Like star
coupler 10, star coupler 12 comprises a slab of dielectric material
24 forming a free space region having two curved, preferably
circular, boundaries 24a and 24b. The array 22 of input waveguides
is connected to the free space region in a substantially uniform
distribution along boundary 24a.
An array 26 of N output waveguides are radially directed from N
output ports toward a focal point F3. The output waveguides in the
array 26 are connected to the free space region 24 in a
substantially uniform distribution along the boundary 24b.
Phase errors caused by mutual coupling between neighboring
waveguides in the arrays 16, 20, 22, and 26 cause increased
crosstalk and reduced efficiency of power transfer in a device such
as the device of FIG. 1. Accordingly, the focal points F1-F4 are
located in the specific locations to minimize such phase errors.
Specifically, focal point F1 is located at the phase center S2 of
array 16, F2 is located at the phase center S1 of array 20, F3 is
located at the phase center S4 of array 22, and F4 is located at
the phase center S3 of array 26.
A phase center for an array of waveguides such as those arrays
shown in FIG. 1 may be considered to be the center of a circle
which most closely approximates a locus of points of equal phase
for optical wavefronts emitted by the array when the array is
excited from a particular input waveguide. In arrays such as those
of FIG. 1 having a significant degree of mutual coupling between
waveguides, the phase center generally is located outside the free
space region a distance d away from the boundary of the free space
region. The location of the phase center of an array of radially
directed waveguides may be determined using the well known
propagating beam method of computing the amplitudes and phases of
radiation flowing from waveguides for any given excitation as a
function of distance from the waveguides. Preferably, it is assumed
that a central waveguide of one of the arrays is excited in the
apparatus in FIG. 1. Assuming input excitation is applied to the
central waveguide, namely, the waveguide directed through the focal
point of the other array of waveguides connected to the same star
coupler, the distance d is selected so as to minimize the variation
in computed phase along some reference circle centered on that
focal point. Various strategies may be adopted regarding this
minimization. For example, d may be selected so as to reduce to
zero as closely as possible the phase difference of the central
waveguide and its two adjacent waveguides. Alternatively, one can
select d so that the phase difference between the central and
marginal waveguides is minimized. This alternative can be shown to
minimize in general the peak value of the phase difference in the
entire array.
Once d has been selected in this fashion, there still may be
unacceptable residual phase errors across the array of waveguides.
These may be reduced by appropriately selecting the length l.sub.s
of the arms of the grating, which results in a grating having a
nonconstant length difference l.sub.s -l.sub.s-1 throughout the
grating.
The apparatus of FIG. 1 can be used as a switch, a multiplexer, or
a demultiplexer. If optical power at a particular wavelength
.lambda..sub.1 is input to a particular input waveguide or input
port in the array 16, the optical input power spreads in the free
space region 18 and is distributed to the M waveguides in the
grating 14 so that portions of the input optical power travel
through each of the M waveguides. Those portions of the input
optical power combine in free space region 24 in such a way that as
much as possible of that power is concentrated on a desired point
along the boundary 24b. This point is selected to be at a desired
input of an output waveguide in the array 26 and thereby the input
optical power is launched into that selected output waveguide. The
location of this concentration of power, and the identity of the
output waveguide to which input optical power is directed, is a
function of the wavelength of input optical power. Thus, one can
select which output waveguide the input power is directed toward by
appropriately setting the wavelength of the input power. The
identity of the output waveguide to which input power is directed
is also a function of the identity of the input waveguide to which
that input power is directed. The apparatus of FIG. 1 can thus
switch input optical power from any of the N input waveguides to
any of the N output waveguides in the case of a device having N
input ports and N output ports.
Notice that the transmission coefficient of the apparatus of FIG. 1
is essentially a periodic function of input wavelength and, in a
particular period, it has a single peak of transmission close to
unity, produced from a particular input port to a particular output
port. If the input and output waveguides are spaced arbitrarily,
the apparatus of FIG. 1 is in general characterized in each period
by N.sup.2 distinct wavelengths of maximum transmission, since
N.sup.2 is the total number of transmission coefficients
corresponding to the various input and output ports. The
differences between these wavelengths are determined by the spacing
of the input and output waveguides. It is important, for most
applications, to choose uniform spacing, so as to cause the above
N.sup.2 wavelengths to essentially coalesce into N wavelengths
.lambda..sub.1 . . . , .lambda..sub.N of maximum transmission in a
particular period. In the following, the device will be assumed to
be arranged in this preferred configuration.
If optical input power comprising a plurality of appropriate
wavelengths .lambda..sub.1, .lambda..sub.2, . . . .lambda..sub.N of
maximum transmission is introduced simultaneously in one of the
input waveguides, each of the wavelengths spreads in free space
region of star coupler 10. Portions of the optical power are
transmitted through the M waveguides of the grating 14 which then
are combined in the free space region of star coupler 12. Optical
power of each wavelength is concentrated at the inputs of different
output waveguides. Each of the wavelengths of optical input power
directed to a single input port is output by the device at
different output ports. The device thus can act as a demultiplexer
of the plurality wavelengths appearing on one of the input
waveguides. The order in which wavelengths appear on the output
waveguides is a function of which of the input waveguides carries
the plurality of input wavelengths. The order thus can be different
when the input wavelengths are directed to different input
waveguide. The device of FIG. 1 thus may be used as an N.times.N
demultiplexer in the case of a device having N input ports and N
output ports.
As mentioned above, the device of FIG. 1 is symmetrical. Therefore,
if optical input power at one of a plurality of different input
wavelengths .lambda..sub.1, .lambda..sub.2 . . . .lambda..sub.N is
applied to each of the input waveguides, all of the wavelengths can
be directed to a single output waveguide. The identity of the
output waveguides is a function of the spatial order in which the
input wavelengths are applied to the input waveguides and also a
function of the magnitude of the wavelengths. The apparatus of FIG.
1 thus may be used as an N.times.N multiplexer in the case of a
device having N input ports and N output ports.
A wavelength-selective N.times.N integrated multiplexer which can
simultaneously multiplex and demultiplex a large number of input
and output channels as illustrated in FIGS. 1 and 2 is particularly
suitable for realization in integrated form by using SiO.sub.2 /Si
technology. Specifically, the waveguides, star couplers, and
optical grating comprise SiO.sub.2 regions patterned on an Si
substrate, for example, by photolithographic techniques.
Typically, the refractive index difference .DELTA.n between the
core and the cladding of each waveguide varies between 0.25 and
0.5%, but the effective difference .DELTA.n.sub.e is smaller, being
between 0.17 and 0.35%. The reason for the smaller values is that
the mode propagates partly outside the core region, and this is
found to reduce the effective refractive difference. For a
waveguide of large width, the reduction factor is determined by the
fraction of the total power that propagates in the core relative to
the total power, and this also gives approximately the reduction
factor for a waveguide of finite width. Accurate procedures for
determining .DELTA.n.sub.e, taking into account the actual
waveguide geometry, are well known, as shown for instance in
"Guided-Wave Optoelectronics", edited by Tamir, Chapter 2 by H.
Kogelnik, 1988, published by Springer Verlag.
In prior art devices of the type illustrated in FIGS. 1 and 2, a
grating formed with SiO.sub.2 /Si technology for typical values of
.DELTA.n of 0.25 and 0.5% supports planar waveguides having a
radius of curvature which is about 10 mm or greater. A radius of
curvature of less than 10 mm results in excessive signal loss at
the bend. Additionally, the limitation on the minimum radius of
curvature of the planar waveguides of the grating determines, to a
large degree, the overall length of the optical grating and,
therefore, the minimum spacing between the star couplers.
By contrast, a planar waveguide embodying the principles of the
invention can have a low loss radius of curvature which is as small
as 6 mm and increased tolerance to fabrication defects.
The new improved bends are realized by ensuring that the width and
the curvature of the planar waveguide meet particular criteria. In
particular, these parameters must be large enough to cause the
fundamental mode of the optical signal to propagate away from the
inner edge of the bend, thus causing negligible illumination of
this edge. In addition, the curvature should be sufficiently large
to effectively cut off higher order modes.
Referring to FIG. 3, there is illustrated one-half of a grating
particularly suitable for use with N.times.N star couplers in
accordance with the principles of the invention. The grating
illustrated in FIG. 3 consists of an array of curved waveguides of
different lengths positioned between two planar free-space regions
formed by dielectric slabs 18, 24. At the circular boundary of slab
18, the waveguides are closely spaced, whereas they are widely
spaced in the central region close to the symmetry axis. In this
region, they are strongly curved, where each waveguide has
approximately the same radius "R" of curvature.
The order of the grating is determined by the difference in length
l between adjacent waveguides. More precisely, it is given by the
relationship ##EQU1## where .lambda. is wavelength; S is the
average spacing between two adjacent waveguides in the central
regions of the grating; and L is the average length over which the
two waveguides are effectively spaced by S.
For many applications, the order of the grating must be large, for
instance, greater than 50. At an order value of 50, the grating
becomes relatively large and is difficult to realize with
satisfactory accuracy, particularly when very low levels of
cross-talk is a requirement. In practice, defects of fabrication
will, in general, cause waveguide width variations that will affect
the propagation constant in each arm of the grating, thus causing
phase errors that will substantially increase cross-talk in a
multiplexer.
With this invention, these phase errors are reduced substantially
by reducing the dependence of the propagation constant on the
waveguide width. Additionally, with this invention the bend losses
are reduced for a given R. Moreover, with this invention, the bend
radius of the waveguide can be substantially reduced without
increasing bend losses.
The smallest radius that can be chosen for a bend without causing
appreciable loss is determined by the effective refractive
difference .DELTA.n.sub.e between the core and cladding. For silica
waveguides with .DELTA.n.perspectiveto.0.25%, for example, the
minimum radius is typically close to 50 mm. See, for example,
Electronics Letters, Jan. 18, 1990, Vol. 26, No. 2, "Broadband
Silica-Based Optical Waveguide Coupler With Asymmetric Structure"
by A. Takagi et al. on pages 132-133.
With the invention here disclosed, this minimum radius of about 50
mm can be reduced by about a factor 1.6 without increasing bend
loss. In general, the loss in a straight waveguide can be
substantially decreased by increasing the width of the waveguide.
But, the width cannot be arbitrarily large. To prevent unwanted
modes, the width of the straight waveguide must be smaller than a
critical value required to ensure that only the fundamental mode
propagates. But, at this width, both side edges of the waveguide
are strongly illuminated by the fundamental mode and appreciable
losses are caused by scattering from rough edges which occur during
the fabrication process. A similar situation arises when the
waveguide is curved and, for this reason, the bend width is
typically chosen to be smaller than the above noted critical value
used for a straight waveguide.
In this invention, the bend performance of a planar waveguide is
substantially improved by increasing the width of the planar
waveguide as the radius of curvature is decreased. More
specifically, the radius of curvature of the planar waveguide is
fixed to be close to the critical value which corresponds to the
largest loss that can be tolerated for the fundamental mode. At the
same time, the width of the planar waveguide is set to allow the
optical energy in the bend to be concentrated along the outer edge
of the bend. Thus, stated differently, the optical energy in the
bend is displaced away from the inner edge of the bend. The
fundamental mode effectively propagates in the vicinity of the
outer edge of the bend, and its propagation constant becomes
effectively independent of the width of the waveguide. As a result,
width variations of the planar waveguide which are caused by
fabrication errors do not contribute to loss. Moreover, as the
field intensity is small at the inner edge, negligible loss is
caused by scattering from this edge.
Since bend losses are essentially independent of the bend width,
the bend radius can be selected to be close to the critical value
which is determined by the largest loss that can be tolerated for
the fundamental mode.
In prior art bends of planar waveguides, this is not practical
because the loss is strongly dependent on the waveguide width and,
therefore, very precise fabrication is required to ensure that the
critical loss is not exceeded.
A bend in a planar waveguide designed in accordance with the
principles of the invention effectively supports with negligible
loss only the fundamental mode when the waveguide width w is equal
to or greater than: ##EQU2## where n is the refractive index; and
R, the radius of curvature is equal to or smaller than the specific
value ##EQU3## This will cause the lowest order asymmetric mode to
suffer losses in excess of 10 dB/radian in all cases of practical
interest, when the width w is chosen in accordance with
relationship (3). Moreover, the inner edge illumination will be
appreciably lower, by approximately 3dB than the outer edge
illumination. In practice, a larger w should be chosen, if
possible, as this will further reduce the inner edge
illumination.
When the above conditions are satisfied, the fundamental mode is
reasonably small and the bend can be connected, with negligible
loss, to a straight waveguide having a width w.sub.g where ##EQU4##
where k=2(.pi./.lambda.)
.lambda.=wavelength of the waveguide
R=radius of curvature of the waveguide
.DELTA.n.sub.e =effective refractive index difference
Substantially 98 percent of the optical energy in the straight
waveguide will be transferred to the planar waveguide bend provided
the axis of the straight waveguide is properly aligned with the
bend mode by displacing the axis of the straight waveguide from the
outside bend edge as illustrated in FIG. 4. The displacement
.delta..sub.g is given by ##EQU5## and both, w.sub.g and
.delta..sub.g can deviate from the above optimum values by as much
as 10%, without causing substantial decrease in efficiency, as is
well known to those skilled in the art. In practice, in the design
of the arrangement illustrated in FIGS. 1 and 3, the gap t between
adjacent waveguides should be sufficiently wide so that there is
only negligible coupling between their fundamental modes. To obtain
this negligible coupling in the region where the waveguide width w
is greater than 2.8, ##EQU6## where k=2(.pi./.lambda.)
in the region with w being given by the above expression.
It will thus be appreciated that those skilled in the art will be
able to devise numerous arrangements which, although not explicitly
shown or described herein, embody the principles of the invention.
Accordingly, all such alternatives, modifications and variations
which fall within the spirit and broad scope of the appended claims
will be embraced by the principles of the invention.
* * * * *